BZ and the Turing Instability Tamas Bansagi BZ Boot Brandeis

What are these two patterns? Tropical fish Turing patterns in a chemical reaction

Alan Turing’s theory ‘The chemical basis of morphogenesis’ Philosophical Transactions of the Royal Society of London, (Series B, No.641, Vol. 237, 37-72,1952). Kinetics Diffusive Transport + u activator v inhibitor D i diffusion coefficients f, g kinetic rate equations Reaction-Diffusion equations - In a Reaction-Diffusion system, patterns stationary in time and periodic in space may develop if D u ≠ D v. - In the same system, if D u = D v ≥0 u and v tend to a stable uniform steady state. - More precisely: D u < D v (Long range inhibition, short range activation required) Nonlinearity Consider:

Alan Turing’s theory Chemical pre-patterning through diffusion driven instability. Formation and development of embryo Early stage Morphogenesis (development of pattern and form) Positional information template Cell differentiation, migration, shape change

Turing patterns in experiment Living systems: Difficult to identify pre-patterning species (morphogens) Mechanisms are very complicated Chemical systems: Relatively easy to identify species Mechanisms tend to be simpler Seemed easier to find/design systems supporting Turing patterns

Turing patterns in experiment Living systems: Difficult to identify pre-patterning species (morphogens) Mechanisms are very complicated Chemical systems: Relatively easy to identify species Mechanisms tend to be much simpler Seemed easier to find/design systems supporting Turing patterns Reality: first Turing patterns reported in 1990 – Clorite-Iodide-Malonic acid reaction (V. Castets, E. Dulos, J. Boissonade, P. De Kepper, 1990) Examples from Biology: Disposition of feather buds in chick (H. S. Jung, 1998) Hair follicles in mice (S. Sick, S. Reinker, J. Timmer, T. Schlake, 2006) Skin pattern regeneration in zebra fish (M. Yamaguchi, E. Yoshimoto, S. Kondo, 2007)

Turing patterns in the BZ reaction Oregonator model X activator Y inhibitor Z oxidized form of catalyst Oregonator model in dimensionless form

Turing patterns in the BZ reaction 1D Oregonator reaction-diffusion system D x = D y = D z =1

Turing patterns in the BZ reaction 1D Oregonator reaction-diffusion system – Homogeneous perturbation D x =0.01, D y = D z =1

Turing patterns in the BZ reaction 1D Oregonator reaction-diffusion system – Inhomogeneous perturbation D x =0.01, D y = D z =1 It is in the model but how can we “slow down” the activator or “speed up” the inhibitor?

Turing patterns – BZ-AOT system AOT – Aerosol OT - sodium bis(2-ethylhexyl) sulfosuccinate Aqueous BZ chemicals Oil (Octane) Water-in-oil microemulsion Communication between droplets collision (fusion and fission) ~ s time scale (exchange of polar species) nonpolar species in oil ~ s time scale Role of Br 2 produced in the reaction quickly diffuses in the oil phase its reaction with malonic acid gives bromide ( Y ) (Thorough review – V. K. Vanag and I. R. Epstein, 2008) Long range inhibition R h = 5-20 nm

Turing patterns – BZ-AOT system 2D 3D Experiments (oil: cyclooctane) Reconstruction (inverse Radon transform) Reconstructed patterns Numerical results in an Oregonator-based model (T. Bansagi, V. K. Vanag, I. R. Epstein, 2011)